ellenbergvenkatesh - Ellenberg, J. S. and A. Venkatesh....

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Ellenberg , J. S. and A. Venkatesh. ( 2007 ) “Reflection Principles and Bounds for Class Group Torsion , International Mathematics Research Notices , Vol. 2007 , Article ID rnm002 , 18 pages. doi:10.1093/ imrn /rnm002 Reflection Principles and Bounds for Class Group Torsion Jordan S. Ellenberg 1 and Akshay Venkatesh 2 1 Department of Mathematics , University of Wisconsin , Madison and 2 Courant Institute of Mathematical Sciences , New York University , New York Correspondence to be sent to: Akshay Venkatesh , Courant Institute of Mathematical Sciences , New York University , New York. e-mail: [email protected] We introduce a new method to bound ± -torsion in class groups , combining analytic ideas with reflection principles. This gives , in particular , new bounds for the 3-torsion part of class groups in quadratic , cubic and quartic number fields , as well as bounds for certain families of higher degree fields and for higher ± . Conditionally on GRH , we obtain a nontrivial bound for ± -torsion in the class group of a general number field. 1 Introduction The goal of the present article is to exhibit some bounds on the ± -part of the class group of a number field which improve on the trivial bound provided by the order of the en- tire class group. As such , they represent evidence towards a conjecture that the ± -part of the class group of a number field L of fixed degree grows more slowly than any power of the discriminant of L . Such conjectures have been suggested by Duke [ 1 ], for CM fields by Zhang [ 2 , page 10 ] as the “ ± -conjecture , ” and in a stronger form by Brumer and Silverman [ 3 , “Question CL ( ± , d ) ] Proposition 3.4 gives the bound | D | 1 / 3 + ± for the 3-part of the class group of Q ( D ) . This improves the known bounds of [ 4 ] and [ 5 ] and has several corollaries ( cf. [ 4 , Section 4 ]) . In combination with the techniques of [ 4 ] one obtains that there are at most N 0 . 169 ... elliptic curves over Q of conductor N . More directly , it implies that there are ± | D | 1 / 3 + ± cubic extensions of Q with discriminant D . Received July 18 , 2006 ; Revised January 12 , 2007 ; Accepted January 16 , 2007 Communicated by Bjorn Poonen See http://www.oxfordjournals.org/our journals/imrn/for proper citation instructions. c ² The Author 2007. Published by Oxford University Press. All rights reserved.For permissions , please e-mail: [email protected]
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2 J. S. Ellenberg and A. Venkatesh Proposition 3.6 is our most general unconditional result on ± -torsion. A particu- lar case of Proposition 3.6 is a nontrivial bound on the 3-torsion in even degree exten- sions of Q with large Galois group ; but it also has consequences for ±> 3 and entails , e.g. a nontrivial bound for the 5-torsion part of the class group of any quadratic extension of Q ( 5 ) .Finally , in Corollary 3.7 we apply these results to show a nontrivial bound on 3-torsion for cubic and quartic extensions of Q .
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This note was uploaded on 12/03/2011 for the course MATH 8430 taught by Professor Clark during the Fall '11 term at University of Georgia Athens.

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ellenbergvenkatesh - Ellenberg, J. S. and A. Venkatesh....

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